Why January 1 and December 31 share a weekday

Question

Suppose it's a normal (non-leap) year, and January 1 falls on a Monday.

What day of the week is December 31 that same year? And why does the first and last day of a non-leap year always land on the same weekday?

Accepted Answer

December 31 is also a Monday, and in any non-leap year, January 1 and December 31 always share the same weekday. Count the days. A non-leap year has 365 days. December 31 is the 365th day, and January 1 is day 1. The number of days between them (from Jan 1 to Dec 31) is 365 - 1 = 364. Reduce by whole weeks. Weekdays repeat every 7 days, so what matters is 364 \bmod 7: 364 = 7 	imes 52 = 364 \quad\Rightarrow\quad 364 \bmod 7 = 0. Exactly 52 whole weeks separate January 1 from December 31, with no leftover days. So December 31 lands on the same weekday as January 1, here, Monday. The leap-year…

Why January 1 and December 31 share a weekday

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